Ionospheric Joule heating and Poynting flux in quasi-static approximation

Energy flow is an important aspect of magnetosphere-ionosphere coupling. Electromagnetic energy is transported as Poynting flux from the magnetosphere to the ionosphere, where it is dissipated as Joule heating. Recently Richmond derived an "Equipotential Boundary Poynting Flux (EBPF) theorem", that the Poynting flux within a flux tube whose boundary is an equipotential curve is dissipated inside the ionospheric foot point of the flux tube. In this article we study Richmond's EBPF theorem more closely by considering the curl-free and divergence-free parts as well as the Hall and Pedersen parts of the ionospheric current system separately. Our main findings are that i) divergence-free currents are on average dissipationless, ii) the curl-free Pedersen current is responsible for the whole ionospheric Joule heating and iii) pointwise match between vertical Poynting flux and ionospheric Joule heating is broken by gradients of Hall and Pedersen conductances. Results i) and ii) hold when integrated over the whole ionosphere or any area bounded by an equipotential curve. The present study is limited to quasi-static phenomena. The more general topic of electrodynamic Joule heating and Poynting flux, including inductive effects, will be addressed in a future study.